statmod (version 1.4.33)

gauss.quad: Gaussian Quadrature

Description

Calculate nodes and weights for Gaussian quadrature.

Usage

gauss.quad(n,kind="legendre",alpha=0,beta=0)

Arguments

n

number of nodes and weights

kind

kind of Gaussian quadrature, one of "legendre", "chebyshev1", "chebyshev2", "hermite", "jacobi" or "laguerre"

alpha

parameter for Jacobi or Laguerre quadrature, must be greater than -1

beta

parameter for Jacobi quadrature, must be greater than -1

Value

A list containing the components

nodes

vector of values at which to evaluate the function

weights

vector of weights to give the function values

Details

The integral from a to b of w(x)*f(x) is approximated by sum(w*f(x)) where x is the vector of nodes and w is the vector of weights. The approximation is exact if f(x) is a polynomial of order no more than 2n-1. The possible choices for w(x), a and b are as follows:

Legendre quadrature: w(x)=1 on (-1,1).

Chebyshev quadrature of the 1st kind: w(x)=1/sqrt(1-x^2) on (-1,1).

Chebyshev quadrature of the 2nd kind: w(x)=sqrt(1-x^2) on (-1,1).

Hermite quadrature: w(x)=exp(-x^2) on (-Inf,Inf).

Jacobi quadrature: w(x)=(1-x)^alpha*(1+x)^beta on (-1,1). Note that Chebyshev quadrature is a special case of this.

Laguerre quadrature: w(x)=x^alpha*exp(-x) on (0,Inf).

The algorithm used to generated the nodes and weights is explained in Golub and Welsch (1969).

References

Golub, G. H., and Welsch, J. H. (1969). Calculation of Gaussian quadrature rules. Mathematics of Computation 23, 221-230.

Golub, G. H. (1973). Some modified matrix eigenvalue problems. Siam Review 15, 318-334.

Smyth, G. K. (1998). Numerical integration. In: Encyclopedia of Biostatistics, P. Armitage and T. Colton (eds.), Wiley, London, pages 3088-3095. http://www.statsci.org/smyth/pubs/NumericalIntegration-Preprint.pdf

Smyth, G. K. (1998). Polynomial approximation. In: Encyclopedia of Biostatistics, P. Armitage and T. Colton (eds.), Wiley, London, pages 3425-3429. http://www.statsci.org/smyth/pubs/PolyApprox-Preprint.pdf

Stroud, AH, and Secrest, D (1966). Gaussian Quadrature Formulas. Prentice-Hall, Englewood Cliffs, N.J.

See Also

gauss.quad.prob, integrate

Examples

# NOT RUN {
#  mean of gamma distribution with alpha=6
out <- gauss.quad(10,"laguerre",alpha=5)
sum(out$weights * out$nodes) / gamma(6)
# }